Communications in Mathematical Physics

, Volume 354, Issue 3, pp 1205–1244 | Cite as

Renormalized Volume

  • A. Rod Gover
  • Andrew Waldron


We develop a universal distributional calculus for regulated volumes of metrics that are suitably singular along hypersurfaces. When the hypersurface is a conformal infinity we give simple integrated distribution expressions for the divergences and anomaly of the regulated volume functional valid for any choice of regulator. For closed hypersurfaces or conformally compact geometries, methods from a previously developed boundary calculus for conformally compact manifolds can be applied to give explicit holographic formulæ for the divergences and anomaly expressed as hypersurface integrals over local quantities (the method also extends to non-closed hypersurfaces). The resulting anomaly does not depend on any particular choice of regulator, while the regulator dependence of the divergences is precisely captured by these formulæ. Conformal hypersurface invariants can be studied by demanding that the singular metric obey, smoothly and formally to a suitable order, a Yamabe type problem with boundary data along the conformal infinity. We prove that the volume anomaly for these singular Yamabe solutions is a conformally invariant integral of a local Q-curvature that generalizes the Branson Q-curvature by including data of the embedding. In each dimension this canonically defines a higher dimensional generalization of the Willmore energy/rigid string action. Recently, Graham proved that the first variation of the volume anomaly recovers the density obstructing smooth solutions to this singular Yamabe problem; we give a new proof of this result employing our boundary calculus. Physical applications of our results include studies of quantum corrections to entanglement entropies.


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  1. AGMO00.
    Aharony O., Gubser S.S., Maldacena J.M., Ooguri H., Oz Y.: Large N field theories, string theory and gravity. Phys. Rept. 323, 183–386 (2000) arXiv:hep-th/9905111 ADSMathSciNetCrossRefGoogle Scholar
  2. ACF92.
    Andersson L., Chruściel P.T., Friedrich H.: On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein’s field equations. Commun. Math. Phys. 149(3), 587–612 (1992) arXiv:0802.2250 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. AGS14.
    Astaneh A.F., Gibbons G., Solodukhin S.N.: What surface maximizes entanglement entropy?. Phys. Rev. D 90(8), 085021–085031 (2014) arXiv:1407.4719 ADSCrossRefGoogle Scholar
  4. AM10.
    Alexakis S., Mazzeo R.: Renormalized area and properly embedded minimal surfaces in hyperbolic 3-manifolds. Commun. Math. Phys. 297(3), 621–651 (2010) arXiv:0802.2250 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. BEG94.
    Bailey T.N., Eastwood M.G., Gover A.R.: Thomas’s structure bundle for conformal, projective and related structures. Rocky Mt. J. Math. 24(4), 1191–1217 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. B95.
    Branson T.P.: Sharp inequalities, the functional determinant, and the complementary series. Trans. Am. Math. Soc. 347(10), 3671–3742 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. BG08.
    Branson T.P., Gover A.R.: Origins, applications and generalisations of the Q-curvature. Acta Appl. Math. 102, 131–146 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. BG01.
    Branson TP., Gover A.R.: Conformally invariant non-local operators. Pac. J. Math. 201(1), 19–60 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. CSS01.
    Čap A., Slovák J., Souček V.: Bernstein–Gelfand–Gelfand sequences. Ann. Math. 154, 97–113 (2001) arXiv:math/0001164 MathSciNetCrossRefzbMATHGoogle Scholar
  10. CEOY08.
    Chang S.-Y., Eastwood M., Ørsted B., Yang Paul C.: What is Q-curvature?. Acta Appl. Math. 102(2), 119–125 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Che84.
    Cherrier P.: Problèmes de Neumann non linéaires sur les variétés riemanniennes. J. Funct. Anal. 57(2), 154–206 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  12. CG15.
    Curry, S., Gover, A.R.: An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity. In: London Mathematical Society. Lecture Note Series. Cambridge University Press, Cambridge (in press). arXiv:1412.7559
  13. DSS01.
    de Haro S., Solodukhin S.N., Skenderis K.: Holographic reconstruction of space–time and renormalization in the AdS/CFT correspondence. Commun. Math. Phys. 217, 595–622 (2001) arXiv:hep-th/0002230 ADSCrossRefzbMATHGoogle Scholar
  14. DM08.
    Djadli Z., Malchiodi A.: Existence of conformal metrics with constant Q-curvature. Ann. Math. 168, 813–858 (2008) arXiv:math/0410141 MathSciNetCrossRefzbMATHGoogle Scholar
  15. ES97.
    Eastwood M., Slovák J.: Semiholonomic Verma modules. J. Algebra 197(2), 424–448 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. EW14.
    Engelhardt N., Wall A.C.: Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime. JHEP 1501, 073–098 (2015) arXiv:1408.3203 ADSCrossRefGoogle Scholar
  17. FG02.
    Fefferman C., Graham C.R.: Q-curvature and Poincaré metrics. Math. Res. Lett. 9(2-3), 139–151 (2002) arXiv:math/0110271 MathSciNetCrossRefzbMATHGoogle Scholar
  18. GGHW15.
    Glaros, M., Gover, A.R., Halbasch, M., Waldron, A.: Singular Yamabe Problem Willmore Energies. arXiv:1508.01838
  19. Gov07.
    Gover A.R.: Conformal Dirichlet–Neumann maps and Poincaré–Einstein manifolds. SIGMA Symmetry Integr. Geom. Methods Appl. 3, 100–121 (2007) arXiv:0710.2585 zbMATHGoogle Scholar
  20. Gov10.
    Gover, A.R.: Almost Einstein and Poincaré–Einstein manifolds in Riemannian signature. J. Geom. Phys. 60(2), 182–204 (2010). arXiv:0803.3510
  21. GLW15.
    Gover, A.R., Latini, E., Waldron, A.: Poincaré–Einstein holography for forms via conformal geometry in the bulk. Mem. Am. Math. Soc. 235 (1106) (2015) arXiv:1205.3489
  22. GSW08.
    Gover, A.R., Shaukat, A., Waldron, A.: Tractors, mass and Weyl invariance. Nucl. Phys. B 812, 424–455 (2009). arXiv:0812.3364; Weyl invariance and the origins of mass. Phys. Lett. B 675, 93–97 (2009). arXiv:0810.2867
  23. GSS08.
    Gover A.R., Somberg P., Souček V.: Yang–Mills detour complexes and conformal geometry. Commun. Math. Phys. 278, 307–327 (2008) arXiv:math/0606401 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. GW13.
    Gover, A.R., Waldron, A.: Submanifold conformal invariants and a boundary Yamabe problem. In: Conference on Geometrical Analysis-Extended Abstract, CRM Barcelona (2013), arXived as Generalising the Willmore equation: submanifold conformal invariants from a boundary Yamabe problem. arXiv:1407.6742
  25. GW14.
    Gover A.R., Waldron A.: Boundary calculus for conformally compact manifolds. Indiana Univ. Math. J. 63(1), 119–163 (2014) arXiv:1104.2991 MathSciNetCrossRefzbMATHGoogle Scholar
  26. GW15.
    Gover, A.R., Waldron, A.: Conformal hypersurface geometry via a boundary Loewner–Nirenberg–Yamabe problem (2015). arXiv:1506.02723
  27. GW16.
    Gover, A.R., Waldron, A.: Renormalized volumes with boundary. arXiv:1611.08345
  28. GW161.
    Gover, A.R.,Waldron, A.: A calculus for conformal hypersurfaces and new higher Willmore energy functionals. arXiv:1611.04055
  29. Gra00.
    Graham, C.R.: Volume and area renormalizations for conformally compact Einstein metrics. In: Proceedings of the 19th Winter School “Geometry and Physics” (Srní, 1999). Rend. Circ. Mat. Palermo (2) Suppl. No. 63, pp. 31–42 (2000). arXiv:math/9909042
  30. Gra16.
    Graham, C.R.: Volume renormalization for singular Yamabe metrics. arXiv:1606.00069
  31. GH05.
    Graham, C.R., Hirachi, K.: The ambient obstruction tensor and Q-curvature. In: AdS/CFT Correspondence: Einstein metrics and their conformal boundaries. IRMA Lectures in Mathematics and Theoretical Physics, vol. 8, pp. 59–71. European Mathematical Society, Zürich (2005). arXiv:math/0405068
  32. GJMS92.
    Graham C.R., Jenne C.R., Mason L.J., Sparling G.A.J.: Conformally invariant powers of the Laplacian. I. Exist. J. Lond. Math. Soc. (2) 46(3), 557–565 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  33. GJ07.
    Graham C.R., Juhl A.: Holographic formula for Q-curvature. Adv. Math. 216, 841–853 (2007) arXiv:0704.1673 MathSciNetCrossRefzbMATHGoogle Scholar
  34. GL91.
    Graham C.R., Lee J.M.: Einstein metrics with prescribed conformal infinity on the ball. Adv. Math. 87, 186–225 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  35. GW99.
    Graham C.R., Witten E.: Conformal anomaly of submanifold observables in AdS/CFT correspondence. Nucl. Phys. B 546(1–2), 52–64 (1999) arXiv:hep-th/9901021 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. GZ03.
    Graham C.R., Zworski M.: Scattering matrix in conformal geometry. Invent. Math. 152(1), 89–118 (2003) arXiv:math/0109089 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. Gra03.
    Grant, D.: A conformally invariant third order Neumann-type operator for hypersurfaces. Master’s thesis, University of Auckland, Auckland (2003)Google Scholar
  38. Guv05.
    Guven J.: Conformally invariant bending energy for hypersurfaces. J. Phys. A 38(37), 7943–7955 (2005) arXiv:cond-mat/0507320 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. HS98.
    Henningson, M., Skenderis K.: The holographic Weyl anomaly. JHEP 9807, 023 (1998). arXiv:hep-th/9806087
  40. LL51.
    Landau L.D., Lifshitz E.M.: The Classical Theory of Fields, Course of Theoretical Physics Series, Vol. 2, 4th edn. Butterworth-Heinemann, Oxford (1980)Google Scholar
  41. LMP01.
    Lake, K., Musgrave, P., Pollney, D.: GRTensorII Maple and Mathematica Package (2001)
  42. LM13.
    Lewkowycz A., Maldacena J.: Generalized gravitational entropy. JHEP 1308, 090 (2013) arXiv:1304.4926 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. LN74.
    Loewner, C., Nirenberg, L.: Partial differential equations invariant under conformal or projective transformations. In: Contributions to Analysis (A Collection of Papers Dedicated to Lipman Bers), pp. 245–272, Academic Press, New York (1974)Google Scholar
  44. Mal98.
    Maldacena, J.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 231–252 (1998) arXiv:hep-th/9711200
  45. Maz91.
    Mazzeo R.: Regularity for the singular Yamabe problem. Indiana Univ. Math. J. 40(4), 1277–1299 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  46. OF03.
    Osher S., Fedkiw R.: Level Set Methods and Dynamic Implicit Surfaces, Volume 153 of Applied Mathematical Sciences. Springer, New York (2003)CrossRefzbMATHGoogle Scholar
  47. PRR15.
    Perlmutter, E., Rangamani, M., Rota, M.: Positivity, negativity, and entanglement. arXiv:1506.01679
  48. P86.
    Polyakov A.M.: Fine structure of strings. Nucl. Phys. B 268, 406–412 (1986)ADSMathSciNetCrossRefGoogle Scholar
  49. ST02.
    Schwimmer A., Theisen S.: Universal features of holographic anomalies. JHEP 0310, 001–018 (2003) arXiv:hep-th/0309064 ADSMathSciNetCrossRefGoogle Scholar
  50. SW10.
    Shaukat A., Waldron A.: Weyl’s Gauge invariance: conformal geometry, spinors, supersymmetry, and interactions. Nucl. Phys. B 829, 28–47 (2010) arXiv:0911.2477 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. RT06.
    Ryu, S., Takayanagi, T.: Holographic derivation of entanglement entropy from AdS/CFT. Phys. Rev. Lett. 96, 181602–181607 (2006). arXiv:hep-th/0603001; Aspects of holographic entanglement entropy. JHEP 0608, 045–099 (2006) arXiv:hep-th/0605073
  52. Sta05.
    Stafford, R.: Tractor calculus and invariants for conformal sub-manifolds. Master’s thesis, University of Auckland, Auckland (2005)Google Scholar
  53. TW16.
    Taylor M., Woodhead W.: Renormalized entanglement entropy. JHEP 1608, 165–206 (2016) arXiv:1604.06808 ADSMathSciNetCrossRefGoogle Scholar
  54. Vya13.
    Vyatkin, Y.: Manufacturing conformal invariants of hypersurfaces, PhD thesis, University of Auckland, Auckland (2013)Google Scholar
  55. Wal84.
    Wald R.M.: General Relativity. University of Chicago Press, Chicago (2010)zbMATHGoogle Scholar
  56. Wil65.
    Willmore, T.J.: Note on embedded surfaces. An. Şti. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. (N.S.) 11B, 493–496 (1965)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsThe University of AucklandAucklandNew Zealand
  2. 2.Department of MathematicsUniversity of CaliforniaDavisUSA

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